Unicyclic Graphs with Equal Total and Total Outer-Connected Domination Numbers

Abstract

Let \(G = (V, E)\) be a graph without an isolated vertex. A set \(D \subseteq V(G)\) is a total dominating set if \(D\) is dominating and the induced subgraph \(G[D]\) does not contain an isolated vertex. The total domination number of \(G\) is the minimum cardinality of a total dominating set of \(G\). A set \(D \subseteq V(G)\) is a total outer-connected dominating set if \(D\) is total dominating and the induced subgraph \(G[V(G) – D]\) is connected. The total outer-connected domination number of \(G\) is the minimum cardinality of a total outer-connected dominating set of \(G\). We characterize all unicyclic graphs with equal total domination and total outer-connected domination numbers.